SOLUTION: Given a family with four ​children, find the probability of the event. The oldest is a girl and the youngest is a boy​, given that there is at least one boy and at least one g

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Question 1155751: Given a family with four ​children, find the probability of the event.
The oldest is a girl and the youngest is a boy​, given that there is at least one boy and at least one girl.
Found 2 solutions by Edwin McCravy, greenestamps:
Answer by Edwin McCravy(20062)   (Show Source): You can put this solution on YOUR website!
There are 16 cases of a family with 4 children, (no twins)
We eliminate by lining through, the ones that are not given.
So that leaves 14.

First   Second    Third   Youngest    Given?   Success?
Born     Born      Born
  B        B         B        B        No        
  B        B         B        G        Yes        No
  B        B         G        B        Yes        No
  B        B         G        G        Yes        No
  B        G         B        B        Yes        No
  B        G         B        G        Yes        No
  B        G         G        B        Yes        No
  B        G         G        G        Yes        No
  G        B         B        B        Yes        Yes
  G        B         B        G        Yes        Yes        
  G        B         G        B        Yes        Yes
  G        B         G        G        Yes        Yes
  G        G         B        B        Yes        Yes
  G        G         B        G        Yes        Yes
  G        G         G        B        Yes        Yes
  G        G         G        G        No 

So that's 7 successes out of 14, so the probability is 7/14 which
reduces to 1/2.

Edwin
Answer by greenestamps(13206)   (Show Source): You can put this solution on YOUR website!


First consider the probability that with four children the first is a girl and the last is a boy:

P(GxxB) = (1/2)(2/2)(2/2)(1/2) = 4/16

Now consider the requirement that there be at least one boy and at least one girl. That eliminates two possible cases -- all boys or all girls -- so the denominator of the probability fraction is 16-2 = 14.

Of the two cases that were thrown out, neither has a girl first and a boy last, so the numerator of the probability fraction is still 4.

ANSWER: 4/14, or 2/7

Note there are other formal ways of calculating that probability; however, for a small problem like this, it is easiest simply to list all 16 possible sequences of 4 boys or girls and find how many satisfy the given conditions. 14 of the 16 satisfy the condition that there is at least one boy and one girl; and 4 of those 14 have a girl first and a boy last.
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